Consider the pendulum with viscous damping introduced during class: +0+ sin(0) = 0. (1) (a) Define...

 

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Consider the pendulum with viscous damping introduced during class: +0+ sin(0) = 0. (1) (a) Define the state as (0,0). Let V (0,0) = 2 + (+0) +2(1-cos(0)). Show that V is a Lyapunov function for this dynamical system. (b) Use the function V(0,0) to show that the origin (0, 0) is an asymptotically stable equilibrium point. (c) Find a Lyapunov function V (0,8) for this dynamical system, different from V(0,0) and from the one given in class. Consider the pendulum with viscous damping introduced during class: +0+ sin(0) = 0. (1) (a) Define the state as (0,0). Let V (0,0) = 2 + (+0) +2(1-cos(0)). Show that V is a Lyapunov function for this dynamical system. (b) Use the function V(0,0) to show that the origin (0, 0) is an asymptotically stable equilibrium point. (c) Find a Lyapunov function V (0,8) for this dynamical system, different from V(0,0) and from the one given in class.